error analysis of non-iterative friction factor formulas relative to

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error analysis of non-iterative friction factor formulas relative to
Journal of Naval Science and Engineering
2012, Vol.8, No.1, pp.1-13
ERROR ANALYSIS OF NON-ITERATIVE
FRICTION FACTOR FORMULAS RELATIVE
TO COLEBROOK-WHITE EQUATION
FOR THE CALCULATION OF PRESSURE
DROP IN PIPES
Assist.Prof. M. Turhan ÇOBAN
Ege University, School of Engineering,
Department of Mechanical Engineering,
Bornova, Izmir, Turkiye
[email protected]
Abstract
Pressure drop in pipes can be calculated by using Darcy-Weisbach
formula. In order to use this formula, Darcy friction factor should be
known. The best approximation to Darcy friction factor for turbulent flow is
given by Colebrook-White equation. This equation can only be solved by
numerical root finding methods. There are several other approximate
equations to Darcy friction factor with some relative error compared to
Colebrook-White equation. In some of these equations the percentage error
is so small that they can be used directly in place of Colebrook equation. In
this study relative error of several equations are re-evaluated.
BORULARDAKİ SÜRTÜNME KAYIPLARI
ANALİZİNDE DARCY-WEISBACH
SÜRTÜNME KATSAYISI HESAPLARINDA
COLEBROOK-WHITE DENKLEMİ YERİNE
GEÇECEK DÖNGÜSEL OLMAYAN
ÇÖZÜMLÜ DENKLEMLERİN HATA ANALİZİ
Özetçe
Borulardaki sürtünme basınç kayıpları Darcy-Weisbach formula ile
hesaplanır. Bu basınç kaybını hesaplamak için f Darcy sürtünme
1
M. Turhan ÇOBAN
katsayısının hesaplanması gereklidir. Türbülanslı akışlarda Darcy
sürtünme katsayısının hesaplanmasında en geçerli yöntem Colebrook-White
denklemidir, ancak bu denklem sayısal kök bulma yöntemleri kullanılarak
çözülebilen bir denklemdir. Colebrook-White denklemine yaklaşım yapan ve
direk olarak çözülebilen çeşitli denklemler mevcuttur. Bu denklemlerin
bazılarının Colebrook-White denklemiyle kıyaslandığında hata yüzdeleri
çok küçük olduğundan, direk olarak bu denklemin yerine kulanılmaları
mümkündür. Yazımızda çeşitli Darcy sürtünme faktörü denklemlerinin
Colebrook – White denklemine gore göreceli hatası irdelenmiştir.
Keywords: Darcy –Weisbach pressure drop formula, Pressure drop in
pipes, Colebrook equation, Friction factors.
Anahtar Kelimeler: Darcy-Weichbach basınç düşümü, Boru içi basınç
düşümü, Colebrook denklemi.
1.
INTRODUCTION
The pressure loss in pipe flow is calculated by using DarcyWeisbach equation. The equation is given as:
L V2
∆P = f ρ
(1)
D 2
In equation (1) ∆P is the pressure drop, f is Darcy friction factor, D is the
hydraulic diameter of the pipe and V is the average velocity. Darcy friction
factor f depends on the flow regime. For fully developed laminar flow
(Reynolds number Re < 2100) friction factor can be determined from
Hagen-Poiseuille equation as
64
f =
(2)
Re
Where, Re Reynolds number. The definition of the Re number can be given
as:
ρVD
(3)
Re =
µ
2
Error Analysis of Non-Iterative Friction Factor Formulas Relative to ColebrookWhite Equation for The Calculation of Pressure Drop in Pipes
Where ρ is the density and µ is the dynamic viscosity of the fluid. In
equation (2) the friction factor changes inversely with Reynolds number.
For transition region (2100 ≤ Re ≤ 4000) and turbulent region (Re ≥ 4000) in
smooth as well as rough pipe the friction factor can be described by
Colebrook-White equation.
2.
ANALYSIS
FRICTION FACTOR EQUATIONS AND ERROR
Colebrook-White equation (1937)[4]
Colebrook-White equation can be defined as
 (ε / D)
1
2.51 
= −2 log10 
+

f
Re f 
 3.7
(4)
Where є/D is the relative roughness which is the ratio of the mean height of
roughness of the pipe to the pipe diameter. As seen from equation (4) the
friction factor is function of Reynolds number and pipe roughness (є).
Colebrook-White equation cannot be solved directly due to it’s an implicit
form as the value of f appears on both side of the equation. In order to solve
equation (4) a numerical root finding method, for example Newton-Raphson
method can be used.
Assume
1
(5)
X =
f
 (ε / D) 2.51 
f ( X ) = X + 2 log10 
+
X
Re 
 3 .7
2 . 51
df ( X )
Re
=1+ 2
dX
 ( ε / D ) 2 . 51 
 3 . 7 + Re X 


3
(6)
(7)
M. Turhan ÇOBAN
X k +1 = X k −
[ f ( X )]k
k = 0,......, n
(8)
 df ( X ) 
 dX 
k
Equation (8) is required to be solved iteratively. In order to solve the
equation, an initial guess is needed. If the value of the first guess is diverge
from the true root value too much equation might converge very slowly or
might not converge at all. In order to find the first guess value one of the
approximate formulas given below can be used. For example Haaland
equation can be used to give the first estimation value for the ColebrookWhite equation. The result of some of the approximation equations listed is
very close to the result obtained from Colebrook-White equation. When the
error level of the new equation is relatively small, requirement of using
Colebrook-White equation can be eliminated all together. Calculation of
pipe pressure drop usually calculated by computers. Replacing an iterative
root finding problem with a directly calculatable equation could save
computer calculation time. The simler equation can be easily computed by
using simpler computational environment such as programmable calculators
or spreadsheet programs such as MS Excel. Some of the Colebrook-White
equation approximation formulas are listed below.
Haaland equation (1983) [23]
A good approximate equation shown. Haaland equation valid for turbulent
flow (Re>2300)
  (ε / D )  1 .11 6 . 9 
1
= − 1 . 8 log 10  
 +

Re 
f
  3 . 7 
(9)
Moody equation (1944) [9]
Developed a relationship that is valid for all ranges of Reynolds numbers
and relative roughness as follows.
4
Error Analysis of Non-Iterative Friction Factor Formulas Relative to ColebrookWhite Equation for The Calculation of Pressure Drop in Pipes
f = 5 . 5 x10
−3
 
10 6
1 +  2 x10 4 (ε / D ) +
Re
 



1/ 3



(10)
Wood equation (1966) [18]
Its validation region for Re >10000 , and 10-5< (ε / D ) < 0.04
f = a + b Re −C
Where
(11)
a = 0.53(ε / D ) + 0.094(ε / D )
(12)
b = 88(ε / D )
(13)
0.225
0.44
C = 1.62(ε / D )
0.134
(14)
Churchill equation (1977) [3]
This equation is valid for all ranges of Reynolds numbers.
 8  12

−3 / 2
f = 8 
 + ( A + B)

 Re 

1 / 12
(15)
Where

 (ε / D )  7  0 . 9  
A =  − 2 log 10 
+
  

3
.
7
Re

  


16
(16)
16
 37530 
B=

 Re 
(17)
Chen equation (1979) [2]
Chen proposed the following equation for friction factor covering all the
ranges of Reynolds number and relative roughness.
5
M. Turhan ÇOBAN
 (ε / D)  5.0452 A 
1
= −2 log10 
−
Re 
f
 3.7065 
(18)
 (ε / D)1.1098 5.8506 
A = log10 
+ 0.8981 
Re
 2.8257

(19)
Where
Swamee-Jain equation (1976) [14]
Its validation region for 5000 >Re>107, 0.00004< (ε / D ) <0.05. SwameeJain have developed the following equation to the Darcy friction factor.
f =
0.25
A2
(20)
Where
 (ε / D) 5.74 
A = log10 
+ 0.9 
Re 
 3 .7
(21)
Zigrang - Sylvester equation (1982) [20]
Developed a relationship that is valid for all ranges of Reynolds numbers
and relative roughness as follows
 (ε / D)  5.02 B 
1
= −2 log10 
−
Re 
f
 3.7 
 (ε / D) 13 
A = log10 
+

Re 
 3 .7
 (ε / D) 5.02 A 
B = log10 
−

Re 
 3 .7
6
(22)
(23)
(24)
Error Analysis of Non-Iterative Friction Factor Formulas Relative to ColebrookWhite Equation for The Calculation of Pressure Drop in Pipes
Serghides equation (1984) [22]
Serghides equation is an approximation of the implicit Colebrook–White
equation. It is valid for all ranges of Reynolds numbers and relative
roughness as follows.
1
( B − A) 2
= A−
(25)
C − 2B + A
f
Where
 (ε / D)  12 
A = −2 log10 
+

 3.7  Re 
 (ε / D)  2.51A 
B = −2 log10 
+
Re 
 3.7 
 (ε / D)  2.51B 
C = −2 log10 
+
Re 
 3.7 
(26)
(27)
(28)
Goudar- Sonnad equation (2008)[21]
Goudar-Sonnad equation is an approximation of the implicit Colebrook–
White equation. This equation is valid for all ranges of Reynolds numbers
and relative roughness. It has the following form.
 d 

1
= a ln  + δ CFA 
f
 q

(29)
Where
a=
2
ln(10)
7
(30)
M. Turhan ÇOBAN
(ε / D)
3 .7
ln(10)
d=
Re
5.02
s = bd + ln(d )
b=
(31)
(32)
(33)
( s /( s +1))
q=s
g = bd + ln(d q )
q
z =  
g
g
δ LA =
z
g +1

δ CFA = δ LA 1 +

(34)
(35)
(36)
(37)

z/2

( g + 1) + ( z / 3)(2 g − 1) 
2
(38)
Romeo equation (2002) [11]
Developed a relationship that is valid for all ranges of Reynolds numbers
and relative roughness as follows.
 (ε / D)  5.0272 B 
1
= −2 log10 
−
Re 
f
 3.7065 
 (ε / D)  0.9924  5.3326  0.9345 
A = log10 
+



 208.815 + Re 
 7.7918 

 (ε / D) 4.567 A 
B = log10 
−

Re 
 3.827
8
(39)
(40)
(41)
Error Analysis of Non-Iterative Friction Factor Formulas Relative to ColebrookWhite Equation for The Calculation of Pressure Drop in Pipes
In order to determine the relative error of all these approximate formulas,
relative error of each equation is with respect to Colebrook-White equation
is calculated by using the following equation:
Relative Error =
( f Colebrook −White − f ) × 100
f Colebrook −White
(42)
3.
RESULTS
The results for the relative error are shown graphically. Results obtained
from error analysis are briefly explained below.
- In transition region relative error is too high; as the effect of turbulence
increased (for high Reynolds number) the relative error will be decreased.
If the approximation formulas are scaled in the order of relative error, Best
results are obtained from Goudar-Sonnad equation, Serghides equation,
Romeo equation; Ziagrang equation and Chen equation are fallowed up in
the given order.
- When a comparison according to degree of the error is made, GoudarSonnad equation with small percentage error order exceeds 10-9 % is very
close to the result obtained from Colebrook-White equation. Then the next
best equation is Serghides equation with percentage error order 10-5 %
which is also can be used practically.
- Because of these equations is precision enough, the need for use
Colebrook-White iterative solution seems to be eliminated.
9
M. Turhan ÇOBAN
Figure 1. The amount of relative error comparison between Goudar and
Colebrook-White equations
Figure 2. The amount of relative error comparison between Serghides and
Colebrook-White equations
10
Error Analysis of Non-Iterative Friction Factor Formulas Relative to ColebrookWhite Equation for The Calculation of Pressure Drop in Pipes
Figure 3. The amount of relative error comparison between Romeo and
Colebrook-White equations
Figure 4. The amount of relative error comparison between Zigrang and
Colebrook-White equations
11
M. Turhan ÇOBAN
Figure 5. The amount of relative error comparison between Chen and
Colebrook-White equations
Figure 6. The amount of relative error comparison between Haaland and
Colebrook-White equations
12
Error Analysis of Non-Iterative Friction Factor Formulas Relative to ColebrookWhite Equation for The Calculation of Pressure Drop in Pipes
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